How can I stop materials from clipping?
First of all, you should prepare two images A and B such that A flipped horizontally has the same contour (the same alpha channel) as B. Otherwise, you will see that the character changes pose when flipping.
If you do that, then your problem will probably be solved: if you have a a very thin (0.001 m) cube with one big face depicting the front of an astronaut, and the other big face depicting the back of the astronaut, you won't be able to see the wrong side, as the right side will cover the other completely (because the alpha channels match).
Without doing the above, you can enable Backface culling to not display insides of an object:

Also, you don't need a cube and a single plane is enough - just use the Geometry &gt; Backfacing to determine which side you're displaying:

Of course for a single plane, the Backface Culling has to be disabled.